A uniform solid cylinder of mass M and radius R is mounted so that it rotates wi

Question

A uniform solid cylinder of mass M and radius R is mounted so that it rotates with negligible friction on a fixed horizontal axle, as shown below. A light cord is wrapped around the cylinder. The free end of the cord is pulled with a constant tension T = 5 N, giving the cylinder rotational acceleration (starting at rest). a) Find an algebraic expression for the angular acceleration alpha of the cylinder. b) Find an algebraic expression for the angular velocity omega after a length d of the cord is unwound. c) If a mass, m, of weight W= 5 N is hung from the free end of the cord instead of applying a constant tension T, will the angular acceleration of the cylinder be the same (starting at rest)? If not, find an algebraic expression for the angular acceleration alpha.

Explanation / Answer

moment of inertia of cylinder = 0.5 * MR^2

torque on cylinder = force * radius

torque = moment of inertia * acceleration

force * radius = moment of inertia * acceleration

T * R = 0.5 * MR^2 * acceleration

acceleration = 2T / MR

by third equation of motion

v^2 = u^2 + 2as

v^2 = 0 + 2 * a * (d/R)

v = sqrt(2 * 2T * d / MR^2)

v = (2/R) * sqrt(Td/M)

equation of motion of m

mg - T = m(a * R) -------------(1)

equation of motion of cylinder

TR = 0.5 * MR^2 * a -------------(2)

substituting T from equation 1 and 2

a = 2mg / R(M + 2m)

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